离心率e=c/a,c=2,2/a=√2/2,a=2√2,b^2=a^2-c^2=4,
设A（x1,y1),B(x2,y2),
椭圆方程为：x^2/8+y^2/4=1,
y=x+m,代入椭圆方程,
x^2/8+(x+m)^2/4=1,
3x^2+4mx+2m^2-8=0,
根据韦达定理,
x1+x2=-4m/3,
设M（x0,y0),x0=(x1+x2)/2=-2m/3,
y0=-2m/3+m=m/3,
M在圆x^2+y^2=1上,则OP=1,
x0^2+y0^2=1,
4m^2/9+m^2/9=1,
∴m=±3√5/5.